Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(5,6,12,18)(7,19,24,37)(8,25,28,42)(9,15,34,46)(10,17,11,16)(13,23,38,22)(14,29,41,30)(20,47,49,35)(21,43,52,60)(26,39,50,59)(27,53,55,64)(31,36,51,61)(32,45,33,44)(40,57,63,56)(48,65,66,58)(54,62,67,68);(1,2,3,4)(5,7,11,23,12,24,10,22)(6,13,17,37,18,38,16,19)(8,26,30,49,28,50,29,20)(9,31,33,52,34,51,32,21)(14,39,25,47,41,59,42,35)(15,43,45,61,46,60,44,36)(27,48,57,67,55,66,56,54)(40,58,64,68,63,65,53,62);(5,8,10,30,12,28,11,29)(6,14,16,25,18,41,17,42)(7,20,22,26,24,49,23,50)(9,27,33,56,34,55,32,57)(13,35,19,39,38,47,37,59)(15,40,45,53,46,63,44,64)(21,48,31,54,52,66,51,67)(36,58,43,62,61,65,60,68);(5,9)(6,15)(7,21)(8,27)(10,33)(11,32)(12,34)(13,36)(14,40)(16,45)(17,44)(18,46)(19,43)(20,48)(22,31)(23,51)(24,52)(25,53)(26,54)(28,55)(29,57)(30,56)(35,58)(37,60)(38,61)(39,62)(41,63)(42,64)(47,65)(49,66)(50,67)(59,68);(8,28)(14,41)(20,49)(25,42)(26,50)(27,55)(29,30)(35,47)(39,59)(40,63)(48,66)(53,64)(54,67)(56,57)(58,65)(62,68)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350491);